Forward Attraction of Nonautonomous Dynamical Systems and Applications to Navier–Stokes Equations

IF 1.7 4区数学 Q2 MATHEMATICS, APPLIED

SIAM Journal on Applied Dynamical Systems Pub Date : 2024-09-12 DOI:10.1137/23m1626384

Hongyong Cui, Rodiak N. Figueroa-López, José A. Langa, Marcelo J. D. Nascimento

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引用次数: 0

Abstract

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2407-2443, September 2024.
Abstract. In this paper we studied the forward dynamics of nonautonomous dynamical systems in terms of forward attractors. We first reviewed the well-known uniform attractor theory, and then by weakening the uniformity of attraction we introduced semiuniform forward attractors and minimal (nonuniform) forward attractors. With these semiuniform attractors, a characterization of the structure of uniform attractors was given: a uniform attractor is composed of two semiuniform attractors and bounded complete trajectories connecting them. As a consequence, the nature of the forward attraction of a dissipative nonautonomous dynamical system was then revealed: the vector field in the distant future of the system determines the (nonuniform) forward asymptotic behavior. A criterion for certain semiuniform attractors to have finite fractal dimension was given and the finite dimensionality of uniform attractors was discussed. Forward attracting time-dependent sets were studied also. A sufficient condition and a necessary condition for a time-dependent set to be forward attracting were given with illustrative counterexamples. Forward attractors of a Navier–Stokes equation with asymptotically vanishing viscosity (with an Euler equation as the limit equation) and with time-dependent forcing were studied as applications.

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非自治动态系统的前向吸引及其在纳维-斯托克斯方程中的应用

SIAM 应用动力系统期刊》，第 23 卷第 3 期，第 2407-2443 页，2024 年 9 月。摘要本文从前向吸引子的角度研究了非自治动力系统的前向动力学。我们首先回顾了著名的均匀吸引子理论，然后通过弱化吸引子的均匀性，引入了半均匀前向吸引子和最小（非均匀）前向吸引子。通过这些半均匀吸引子，我们给出了均匀吸引子结构的特征：一个均匀吸引子由两个半均匀吸引子和连接它们的有界完整轨迹组成。因此，耗散非自发动力系统的前向吸引力的性质也随之被揭示出来：系统遥远未来的矢量场决定了（非均匀）前向渐近行为。给出了某些半均匀吸引子具有有限分形维度的标准，并讨论了均匀吸引子的有限维度。此外，还研究了前向吸引随时间变化的集合。给出了时变集向前吸引的充分条件和必要条件，并举出了反例加以说明。作为应用，研究了具有渐近消失粘度的纳维-斯托克斯方程（以欧拉方程为极限方程）的前向吸引子和与时间相关的强迫。

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来源期刊

SIAM Journal on Applied Dynamical Systems 物理-物理：数学物理

CiteScore

3.60

自引率

4.80%

发文量

审稿时长

6 months

期刊介绍： SIAM Journal on Applied Dynamical Systems (SIADS) publishes research articles on the mathematical analysis and modeling of dynamical systems and its application to the physical, engineering, life, and social sciences. SIADS is published in electronic format only.